# Difference between unique solution and particular solution

• inter060708
In summary, for linear ordinary differential equations, the terms "unique solution" and "particular solution" differ in emphasis. The unique solution is a specific solution that satisfies given initial conditions, while the particular solution is any solution with a specific choice of constants that satisfies the differential equation.
inter060708

## Homework Statement

For 1st order linear ordinary differential equation, how do "unique solution" and "particular solution" differ?

## Homework Equations

if dy/dx = f(x,y) and partial f(x,y) with respect to y are both continuous, then there exists a unique solution within a region R.

## The Attempt at a Solution

Not really a homework question. I am just curious and sort of confused. The way I understand it is particular solution is a subset of unique solution, is that true?

If a given differential equation f(x,y,y',y'',...) = 0, this is called a homogeneous equation and it has what is called a complementary solution.

If the differential equation f(x,y,y',y'',...) = c + f(x,y), this is called a non-homogeneous equation. The solution to these types of equations are known as particular solutions.

If the ODE is linear, then the solution to the equation will be a linear combination of the complementary and the particular solutions.

If there is one solution for a given ODE with given boundary conditions, then that solution is called unique.

The differ only in emphasis. If the general solution to a differential equation is, say, $y= Ae^x+ Be^{-x}$, then "a specific solution" is any solution with a specific choice of A and B. The "unique solution" is a specific solution that satisfies given initial conditions.

For example, if the differential equation is $d^2y/dx^2= y$ then the "general solution' is, as before, $y= Ae^x+ Be^{-x}$ for any constants A and B. A specific solution (notice the use of the indefinite article, "a") might be $y= e^x- 2e^{-x}$ where I have arbitrarily chosen A= 1, B= -2. The unique solution (notice the use of the definite article, "the") to the differential equation, $d^2y/dx^2= y$ with the initial conditions y(0)= 1, y'(0)= 0, is $(1/2)e^x+ (1/2)e^{-x}$.

Of course, the "unique solution" to the given "initial value problem" is one of the infinite number of "specific solutions" to the differential equation.

## What is the difference between a unique solution and a particular solution?

A unique solution is a solution to a problem that is the only possible answer and satisfies all conditions. A particular solution, on the other hand, is one of many possible solutions that satisfies the given conditions.

## When is a unique solution possible?

A unique solution is possible when there are enough constraints or conditions given to determine a single answer. In other words, when there is only one solution that satisfies all the given conditions.

## Can a particular solution also be a unique solution?

Yes, a particular solution can also be a unique solution. This happens when there are multiple solutions to a problem, but only one of them satisfies all the given conditions.

## How do you find a unique solution?

To find a unique solution, you need to have enough equations or conditions to solve for each variable in the problem. This ensures that there is only one set of values that satisfies all the equations or conditions.

## What happens if there is no unique solution?

If there is no unique solution, it means that there are either too few constraints or too many variables in the problem. This results in multiple solutions or no solutions at all.

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